Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{4q^3 + 24q^2 + 32q}{-7q^3 - 35q^2 - 42q}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {4q(q^2 + 6q + 8)} {-7q(q^2 + 5q + 6)} $ $ t = -\dfrac{4q}{7q} \cdot \dfrac{q^2 + 6q + 8}{q^2 + 5q + 6} $ Simplify: $ t = - \dfrac{4}{7} \cdot \dfrac{q^2 + 6q + 8}{q^2 + 5q + 6}$ Since we are dividing by $q$ , we must remember that $q \neq 0$ Next factor the numerator and denominator. $ t = - \dfrac{4}{7} \cdot \dfrac{(q + 2)(q + 4)}{(q + 2)(q + 3)}$ Assuming $q \neq -2$ , we can cancel the $q + 2$ $ t = - \dfrac{4}{7} \cdot \dfrac{q + 4}{q + 3}$ Therefore: $ t = \dfrac{ -4(q + 4)}{ 7(q + 3)}$, $q \neq -2$, $q \neq 0$